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Friday, February 14, 2014

A Typical Day: Math Lit classroom videos

By request, we've videoed some lessons of a Math Lit (MLCS) class. If you're piloting or planning a pilot, this detailed guide through the clips that follows may be helpful for getting a feel for a typical flow of a lesson.

To start, please print the attachment below of lessons 2.5 and 2.6 from our book, Math Lit. They are the instructor's versions of the lesson. Looking through them as you watch Heather teach parts of each lesson will help make sense of the problems students are working and the flow of the content.

In the instructor pages, you will see icons suggesting to the instructor if the activity is group, whole class, or individual. Time estimates for the entire lesson and each part of the lesson are listed under the icons. Notes based on our class tests are provided. You will see Heather often following the suggested format of the lesson, but she also deviates as needed. Sometimes an activity is suggested to be whole class and she will teach it in groups or vice versa. We both do this as needed depending on the personality of the class as well as their mathematical level of understanding. The text is meant to flex for you and your students.

A few notes:

I've tried to provide the best quality video I can, but I'm not a professional videographer. Please turn up the volume on your computer. When several students are discussing, there will be a lot of conversations. It's not always easy to hear all of them.

Also, due to our schedules, I was not able to video one lesson from start to finish. I recorded the end of lesson 2.5 and a good portion of lesson 2.6. It's a lot of time in the classroom and definitely provides many helpful tips on what happens in a classroom. But I will start with the beginning of a lesson, going through 2.6, and end this post with the end of a lesson, which will be from 2.5.

We like to have 24 - 30 students max in a class. Our developmental math class sizes are capped at 24, which is a nice number of students, especially with group work.

We group students for each unit, called cycles, and have them work and sit with their group every day. Groups usually have 3 or 4 students in them. We use a room that allows the desks to be moved near each other. We also use a document camera to display student pages of the book on a screen. We write on the student pages to model what students should be writing when they are taking notes. Their book is a consumable worktext, meaning it is their book and workbook all in one.

How a lesson begins: Explore

Lessons start with some kind of problem or scenario that students explore, usually in their groups. This part of the lesson is known as the Explore. Heather opens Lesson 2.6 Measure Up (book pages # 172 - 173) by explaining the scenario and getting students started with some geometric formulas. You will see throughout this lesson that sometimes she frames the problems they are to work on and other times tells them what to work on without reading the problem for them. This encourages students to read better and analyze a reading critically for information.

Key takeaways:

It's not necessary to tell students the mathematical objectives they will be working on before moving into content. It's ok to let them solve some problems and see where they need more tools. This motivates the need for the theory and the objectives to be addressed. In this lesson, students see the need for understanding the rules behind exponents.

Students are not told which formulas to use to answer the questions. They must work together to make sense of and solve the problems provided. This makes the activity into a problem instead of an exercise.

The atmosphere of the room can be loud when students are working in groups. As long as students are productive, it's ok to have some noise in the room.

Heather moves around the room to monitor the progress, help students who might be stuck, and help students articulate their thought processes. She doesn't give the answers, though. Students are supposed to work as a group to arrive at answers at this point in the lesson.

At the 9 - 10 minute mark, it's hard to hear all of the conversation but Heather is clarifying diameter and radius and the difference between perimeter and area. These concepts have been seen before in the course but students still struggle with using them.

Explore discussion

Next, Heather discusses the answers to the Explore problems as a class.

Key takeaways:

Heather encourages good mathematical practices while showing how to solve the problems. She starts by asking students if they used a picture, which is a useful technique but often unused by students. She also emphasizes the use of units, that we are dealing with quantities, not just numbers. Doing so gives meaning to the units in the answer and leads to fewer students leaving off units in their answers.

Multiple approaches are valued. Some students saw the problem's solution immediately with a picture. Others needed the calculations to make sense of the problems.

The student who noticed four 8's is more than pi 8's articulated a way of looking at the problem that is atypical but very useful. Estimation and mental math were used, showing students that the calculator is not needed all the time. They can improve their mental math skills by practicing them often. We regularly tell students to do certain types of problems in the book without a calculator to encourage this practice.

Heather asks students to explain how their thought process as she works through problems, correcting mistakes as needed. It is helpful to students to hear how others think through a problem.

Transitioning to new content: Discover

Now that students have had a chance to see a problem where more mathematical knowledge is needed, we begin the work of developing the theory. This part of each lesson is usually the longest component and is known as the Discover. We call it that because we don't necessary use lecture during the direct instruction. Students are still working together often on problems, but the focus is to develop new skills, vocabulary, and notation that can be used.

In this clip, Heather helps students make conjectures about exponent rules by having them complete an activity in groups (book pages #173 - 174). She models what to write in the table and then has students work on the remaining questions in the table.

Key takeaways:

It is more time-consuming to have students discover the rules as opposed to giving the rules in a lecture. However, the learning is deeper and the retention is better, lessening time needed for review of concepts in later lessons.

Students are asked to make conjectures. In the first cycle, they learn about inductive and deductive reasoning, conjectures, and counterexamples. They are applying those old ideas in a new context. This spiral approach is used throughout the book. It leads to greater understanding over time. Instead of doing 100 exercises with a skill when it is first learned, students do enough practice to get some confidence and then see many more uses of the skill over time to solidify doing the skill but also knowing when to use it.

When students were quiet, Heather encouraged them to work together. When one student in the group found the answer but the others weren't at that point, she encouraged the student to explain his reasoning to the others. They benefit from his explanation (which may be different than the instructor's explanation) and he benefits by learning how to articulate his thought processes.

Discover discussion
Next, Heather discusses the answers to the Discover problems as a class and continues with some additional problems as a class.

Key takeaways:

Heather does not treat the students as though they have never been in a math class. Most have had 1 - 2 years of algebra prior to this class. She uses that background to preview some algebraic concepts that will be addressed in the upcoming lessons in much more detail.

Heather encourages being careful with language and notation. "Distributing" has a specific mathematical meaning, so using it in other instances can be problematic. She also talks about "cancelling" since this is not a mathematical procedure, just a verb we use often. Mathematically, we divide factors. That precision can make a difference in understanding.

When writing the conjectures, she emphasizes when the rule is used. That will be necessary when students start applying the rules.

Students are more inquisitive and curious about mathematics with this activity-based, problem-solving approach to the course. By constantly looking deeper at concepts, often ones students have seen before, and always asking students "why?", students learn that it's ok to wonder. Those discussions, although something tangential, can be really interesting and illuminating. Heather's class was very interested in the idea of imaginary numbers in lesson 2.4, even though we barely touch on that concept in this course.

One of her students talked to me after class. I had thanked her for being recorded and mentioned I thought she had a nice class. She told me this:

"I've never liked math. I really worried at the beginning about all the word problems. But now I really like this class and how we do things."

Remaining parts of the lesson

At this point, her class was done for the day. Let's look at what she'll do during the next class to finish the lesson.

On page 175, a How It Works box summarizes the rules that the students just discovered. Heather will point out each rule, mentioning the notation for writing the rule and the verbal way of thinking of the rule. At the bottom of the page are some traditional practice problems. She will work through those as a whole class so that students can see which rule is being applied and why it's being used. If students need more practice, we may do a few more problems. However, there is substantial practice on this skill in the MyMathLab homework.

Students will then work through #7 and 8 in groups, allowing them get more skill practice while applying the new skills they've learned in a context. The problems will require students to complete geometric calculations with units in their calculations, something most students have never done. This shows them why the answer is in linear, square, or cubic units. It's also a common practice in science, providing students with a skill they can use in other disciplines. Heather will go through each problem after students have had a little time to work with them.

Next, she will work through the Connect portion of the lesson which allows students to extend their new skills into a more involved problem than the original Explore problem or possibly in a new context. Because these problems in this Connect are difficult, we do them as a whole class.

She will then wrap the lesson up by discussing the Reflect box on page 177 and then remind students to do the MyMathLab homework for skill practice and then all of the book homework to work with application of the skill. This particular book homework assignment is quite short but that is due to the MyMathLab assignment being very lengthy.

I was able to video the end of Heather's Lesson 2.5. Let's look at how a lesson closes in more detail.

Apply new knowledge: Connect
In Lesson 2.5 An Ounce of Prevention (book pages #166 - 172), students work with the concept of the mean of a data set. The calculations are usually the easiest part of the lesson for students. Making sense of how the mean works is the more difficult part, but also very important. The students worked through a grade situation in the Explore, learned about the definition and concepts of the mean in the Discover, including working with physical pieces (or pictures) to get more visuals of the mean. At this point, they are ready to make some connections to what they've learned and the original context: grades.

Key takeaways:

Student success ideas naturally come into the lesson by using a grades scenario for working with means. This provides a chance for a good discussion on the importance of front-loading your grades and not waiting until the end of the semester to improve them. This lesson is taught not long after receiving their first test grades, making it even more relevant to students.

Close the lesson: Reflect and homeworkHeather discussed the point of the lesson, including the title, which is short for the saying "an ounce of prevention is worth a pound of cure." Most students under the age of 25 are not familiar with this saying, surprisingly.

The video ended at this point, but Heather continued with some reminders about homework. She reminded students to do both the online and book homework. This book homework assignment is longer than the one in lesson 2.5 and is more typical of the length of most book homework assignments.

Hopefully this bird's eye view of one of our classes gives you a better sense of how the course works. The lesson flow described here is typical. Students quickly adapt to the approach used.

Current Projects

My main project is creating the Mathematical Literacy for College Students (MLCS) course in IL and beyond. It is similar to the first course in the Carnegie Quantway Initiative. This course is for a student placing into beginning algebra who is a non-STEM major. It incorporates numeracy, proportional reasoning, algebraic reasoning, functions, geometry, and statistics along with critical thinking, reading, writing, and problem solving. I worked with a colleague, Heather Foes, to pilot and co-author a book, Math Lit, which is published by Pearson.

I am a member of AMATYC's New Life for Developmental Math that worked with the Carnegie Foundation to create the course and its objectives. I took the objectives and modified them slightly to be more acceptable in IL and started the process of bringing the course to life in IL. Additionally, I'm working with faculty throughout the country to modify the course to make it work for their state requirements and college needs. If you are interested in this course or some derivative of it, please email me.

We have a successful developmental math redesign (documents available below) but felt not all students were served. The MLCS course adds another layer to the redesign and supports students who are not STEM-bound.